# Family of curves Parabolic cluster bundles in and tufts in${\ displaystyle f_ {a} (x) = ax ^ {2}}$ ${\ displaystyle (0,0)}$ ${\ displaystyle a}$ A family of curves , also a family of functions , a family of functions or a parameter function , is a set of different curves whose mapping rules differ in at least one parameter . Special cases are the tuft , a one-parameter group, and the bundle , a group with a point common to all functions.

## definition

The family is a set of points on a curve, curves on a surface or surfaces in space , each of which is described by an equation or a system of equations with variable parameters.

According to another definition , a family of curves results from the graph of a function, in which a free parameter of the relevant function is varied in a parametric representation .

Dynamic geometry systems are particularly suitable for visualizing functional groups .

## special cases

• If all the diagrams of the family of functions are straight lines , one speaks of a family of straight lines.
• If the individual straight lines also run parallel, they are called a set of parallels .
• If all the straight lines involved intersect at one point, it is a straight line bundle .
• If all the straight lines involved both intersect at a point and lie in a plane , it is a straight line bundle .
• If all curves of the family are parabolas , one speaks of a parabola family.

## Examples

• all curves of the family of curves belonging to the function run parallel to the axis (straight line). The parameter of this family of curves is .${\ displaystyle \! \ f_ {p} (x) = p}$ ${\ displaystyle x}$ ${\ displaystyle p}$ • all curves of the family belonging to the function are parabolas through the coordinate origin (see illustration). The parameter is .${\ displaystyle f_ {a} (x) = ax ^ {2}, \, a \ neq 0}$ ${\ displaystyle a}$ • all curves of the family belonging to the relation are concentric circles . The parameter is here .${\ displaystyle \! \ r ^ {2} = x ^ {2} + y ^ {2} \ \ Leftrightarrow {} \ r = {\ sqrt {x ^ {2} + y ^ {2}}}}$ ${\ displaystyle r}$ ## literature

• Set of curves In: Schülerduden - Mathematics II . Bibliographisches Institut & FA Brockhaus, 2004, ISBN 3-411-04275-3 , pp. 241–242
• Mark Yes. Vygodskij: Higher mathematics at hand: definitions, theorems, examples . Springer 2013, ISBN 9783322901132 , p. 696